Question

Annual profit (P) is the product of total annual sales (S) and profit per unit sold (X) that is ? = ? × ?. It is desired to know the probability distribution of the random variable P when X and S have the following assumed probability mass functions (X and S are independent):

a) (3 points) Calculate the expected Annual Profit.

b) (3 points) What is the standard deviation of the Annual Profit?

X(unit profit)	X(unit profit)	S(annual sales)	S(annual sales
value	probability	value	probability
0.2	0.07	1780	0.1
0.3	0.43	3800	0.4
0.6	0.35	5800	0.35
1	0.15	14000	0.15

Answer 1

First,we need to find the probability distrust ofAnnual profit (P)

Since probability distribution of total annual sales(S) and profits per unit sold (X) are independent therefore. Probability of P = Probability of S * Probability of X

P(Annual Sales). Annual sales Probability

356. 0.007

1140. 0.172

3480. 0.1225

14000. 0.0225

a) expected annual profit = Annual sales * Annual sales probability

0.007*356+0.172*1140+0.1225*3480+0.0225*14000=940(approx)

  

Lets now find square of deviation from expected annual profit for each distribution of annual profit

squared deviation

341056

40000

6451600

170563600

b) Standard deviations of expected profit = ( squared deviation * Probbility of annual profit)

( 4637270)=2153.43